Hidden Line Removal

1
Hidden Line Removal

• Applying vector algebra to the problem of
  removing hidden lines from wire-frame models

2
Convex objects

• We first focus on modeling convex objects
• One outward face cannot hide another
• So visible faces can be drawn in any order
• Examples barn, cube, and dodecahedron

3
A non-convex object
part of this face is hidden by that face
eye of viewer
4
2D Polygons in 3D space

• A polygon is a 2-dimensional figure
• Its edges and corner-points are coplanar
• Its a bounded region of an infinite plane
• Any plane in 3-space can be described by an
  first-degree (linear) algebraic equation ax
  by cz d
• Alternatively, using vector algebra, a plane can
  be described using a reference-point Q and a
  direction-vector N (a, b, c), as N QP 0

5
Face-Planes of solid objects

• Any plane surface has two sides
• When a plane is a surface of a solid object it
  has an inside surface and an outside surface
  (a viewer sees the outside one)

front
back
viewer
6
Angles and cosines

• An angle of 90-degrees is a right angle
• Angles less than 90-degrees are acute
• And angles over 90-degrees are obtuse

obtuse angle
right angle
acute angle
cosine lt 0
cosine gt 0
cosine 0
7
Vectors and dot-products

• Vectors u (ux, uy, uz) and v (vx, vy, vz)
  have a dot-product
• uv ux vx uy vy uz vz
• A vectors length u equals sqrt( uu )
• A dot-product is related to the cosine of the
  angle ? between the two vectors uv
  uvcosine(?)
• So sign of dot-product tells angles type

8
Visibility of face-planes
N outward pointing normal vector
N
D
?
DN gt 0 (acute angle ?)
outer surface is visible
D direction vector (from plane toward viewers
eye)
9
Hidden face-plane
N outward pointing normal vector
N
?
D
DN lt 0 (obtuse angle ?)
outer surface is hidden
D direction vector (from plane toward viewers
eye)
10
Format of models data-set

• Added data needed to describe our model
• Number and Location of vertices as before
• Edge-list is no longer needed (zero edges)
• Face-list is the new information to be add
• Each face is a polygon number of sides, list of
  vertices in counter-clockwise order (as viewed
  from the outside of the model), and the
  face-color to be used for the face

11
Example data-set cube
7
0
Face-List (6 faces)
3
4
4-sided 0, 1, 2, 3 (blue) 4-sided 1, 0,
7, 6 (green) 4-sided 2, 5, 4, 3
(cyan) 4-sided 4, 5, 6, 7 (red) 4-sided
6, 5, 2, 1 (magenta)
6
1
5
2
The vertices of each face should be listed in
counterclockwise order (an
seen from the outside of the cube)
12
Why counterclockwise order?

• We need to compute the outward-pointing normal
  vector for each polygonal face (to determine if
  that outward face is visible)
• That normal vector is easily computed (as a
  vector cross-product) if the vertices were listed
  in counterclockwise order

13
Recall the cross-product uv

• If u ( ux, uy, uz ) and v ( vx, vy, vz ),
  then w uv is given by these formulas wx
  uyvz uzvy wy uzvx uxvz wz uxvy
  uyvx
• Significance cross-product w makes a
  90-degree angle with both u and v (so its
  normal to a plane containing u and v)

14
Three consecutive vertices
pq
p v1 v0 q v2 v1
V1
q
p
V2
V0
pq will be the outward-pointing normal
vector (if v0, v1, v2 occurred in
counterclockwise order)
15
Demo program

• Our filldemo.cpp application reads in the data
  for a 3D model, determines which of its polygonal
  faces are visible to a viewer, fills each visible
  face in its specified color, and then draws the
  edges of visible faces
• Datasets plato.dat, redbarn.dat, plane.dat
• Also two models for a non-convex object
  corner1.dat and corner2.dat (same object)

16
In-class exercises

• Try to create the data-sets for some more
  interesting 3D objects, (by writing a C
  program to generate the objects vertices and the
  face-lists)
• Example An octagonal prism
• Divide a circle into eight equal-size angles
• Use sine and cosine to locate upper vertices
• Use sine and cosine to locate lower vertices
• Use number-patterns to generate its ten
  face-planes